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Spontaneous Symmetry Breaking and Goldstone Bosons – Applications in a Broad Range of Physical Systems

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Spontaneous Symmetry Breaking and Goldstone Bosons – Applications in a Broad Range of Physical Systems Spontaneous Symmetry Breaking and Goldstone Bosons – Applications in a Broad Range of Physical Systems By Soon-Yong, Chang Abstract: The concept of Goldstone bosons that arises as consequence of the breaking of continuous symmetry was born in the high energy physics and found interesting applications in other fields of physics such as solid state. This is a review assay of the reference articles where a summary of basic theory and some interesting applications are outlined. 1. MOTIVATION We can argue that if there is to be a general theory of physics, it should be capable of accounting for phenomena occurring at different energy scales: low(solid state physics), middle(nuclear physics) and high(particle physics). We know that quantum mechanics is the basis, however the techniques that are specific to each field have been on the divergent paths from others that made difficult to attain an integrate picture of matter as a whole. However, there is a language that is applicable to all of the energy ranges: that of the field theory either classical or quantum mechanical, that exploits the properties of symmetry and the simplification of the problem by low-energy expansion. It is in this context that the theory of Goldstone bosons acquires its powerful application. In short, Goldstone bosons are massless degrees of freedom that appear as consequence of the breaking of the continuous symmetry of the ground state. These modes or ‘particles’ become decoupled in the low energy limit and exhibit the following additional properties: 1). Goldstone particles can be represented by fields that are rotational scalars, hence these particles are spinless bosons. 2). There is precisely one Goldstone particle for each symmetry generator that is broken. In this essay, I will try to summarize the underlying basic theoretical concepts and some physical examples; starting with pion-pion scattering in the high energy physics where this whole approach first found its application and experimental verification and then discuss about some relevant and on-going topics of the solid state physics. 2. THEORY 2.1. BACKGROUND We can simplify a great deal of problem by considering the lower energy (or temperature) limit of a physical system. There are fewer degrees of freedom and the interactions between the states become less important in many cases. A generally accepted method of studying low energy systems is to formulate ‘effective lagrangians’. The formulation of the problem in the language of the field theory presupposed the identification of the relevant low-energy degrees of freedom and symmetries. It seem like the choice of the variables is itself a critical part of the problem. To supplement the effectiveness of this approach there are power laws that allows simple identification of the interaction terms. First of all, two important theorems should be remarked, as this whole approach is based on them: +The first one is Noether’s theorem: This theorem states that there is a conserved current j µ whenever the action is globally symmetric. Introducing the language of the field theory the = 4 φ ∂ φ φ action is S ∫ d xL( , µ ), where (x) denotes the fields relevant to the problem. Let’s say δφ =ξ φ ω a that the fields transform according to a ( ) , then the invariance of S implies the δ = ∂ ω a µ transformation of the lagrangian density at most by a total derivative: L µ ( Va ) for µ φ µ ≡ − ∂L ξ + µ some quantity Va ( ) . Noether’s theorem states that if we define ja ∂(∂µφ) a Va , this ∂ µ = quantity satisfies the relation: µ ja 0 . The last relation is valid for relativistic as well as non-relativistic systems. It is easy to notice that this is analogous to the usual continuity ≡ r 0 equation. Thus we can define as ‘charge’ Qa (t) ∫ dxja (x) . The existence of such a ‘current’ that obeys the conservation law gives rise to the following theorem. +Goldstone’s theorem: When a symmetry is broken (by a system’s ground state), weakly coupled states known as Goldstone bosons appear. If this state is denoted by | G > , then the matrix element of the density of charge < Ω| j 0 | G > cannot vanish; where | Ω > is the ground state of the system. The proof is based on the property that the charge as defined by Noether’s theorem is the generator the symmetry transformation (δφ = i[Q,φ(x)]) and the fact that the field φ(x) must have a nonzero expectation value in the ground state: < Ω | φ(x) | Ω >≠ 0 . The important consequences of this theorem are: i). The Goldstone boson must be ‘gapless’, that is when the momentum vector goes to zero the energy must also go to the zero limit. We can easily see that this is equivalent to the masslessness of the Goldstone particle in the relativistic systems: E( p) = p 2 + m2 . ii). In the case of the exact symmetry, the Goldstone bosons become completely decoupled of all the interactions in the limit that their momentum goes to zero. In many real cases, we don’t have ‘an exact symmetry’. However, we can treat the system as basically symmetric with some perturbations that account for the violations of the above- mentioned properties. The approximate symmetry particles are known as pseudo-Goldstone bosons, which can have light mass and weak interactions. In general, the lagrangian does not show explicitly the property that the couplings become weaker for the limit of vanishing momentum. This is in fact hidden from the simple identification and becomes apparent once the scattering amplitude is calculated. We can adopt two postures in such a case: to keep the lagrangian as original which displays the renormalizability or to make a change of variables that manifest in the lagrangian the Goldstone modes; the latter at the expense of losing the renormalizability. In the context of the current discussion, this last choice is the preferred one. We have to remember that the symmetry transformations are mathematical in nature and do not depend on the details of the actual physical model, thus the generality of the method. The symmetry operations that play role in Noether’s theorem and hence also in Goldstone’s theorem belong to the so called continuous symmetry group; that is, the group elements can be parametrized by a continuous parameter (as the angle for the operation of rotation) as opposed to a discrete label. These groups obey Lie algebra and, not surprisingly, are called Lie groups. The properties of Lie algebra are well known mathematically of which we can denote that: i). It allows representation as finite-dimensional unitary matrices. ii).There exist the so-called generators Ta , which are finite-dimensional and hermitian matrices. These matrices obey a defining commutation relation. A symmetry operation can be = α a written as: g exp[i Ta ]. 2.2. GENERAL APPROACH Let λ be a parameter that gives the strength of the interaction potential, we can take a semi- classical approach for the limit λ <<1. In this context we find the minimum (v ) of the scalar potential and suppose that the low-degrees of freedom are the small harmonic oscillations of the field about the minimum of the scalar potential. If we define the real and imaginary parts of the field fluctuations as R ≡ 2 Re(φ − v) and I ≡ 2 Im(φ) , we can expand the scalar potential in powers of R and I. From this expansion it becomes apparent the corresponding Feynmann diagrams of different order and contributions to the lagrangian. 3. EXAMPLES 3.1. APPLICATIONS IN HIGH ENERGY (PIONS) Let’s remember that modern description of the strong interactions is based on the theory of interaction of spin-half quarks and spin-one gluons. This is the so called Quantum Chromodynamics(QCD). This theory is described by the lagrangian density: 1 a µν µ L = − Gµν G − ∑q (γ Dµ + m )q QCD a n qn n 4 n a where Gµ is the field strength tensor for the gluon fields and a =1 ~ 8 labels the generators of the gauge symmetry group of the theory. The quarks are described by qn where n = 1~6; that is there are six different kinds(flavors) of quarks, and γ µ are the Gell-Mann matrices. It is this kind of lagrangians that give rise to the bound states (hadrons) such as pions (π ) that are composed of a pair of quarks. In the limit of vanishing masses for the constituent quarks we have the exact symmetry of QCD known as ‘chiral symmetry’; so called because it treats left handed fermions (quarks) differently from the right handed fermions. As the chiral symmetry is only approximate, we have a better agreement of the theory at light mass cases; that is when the building quark masses are much lighter than the scale of the strong interactions. This is precisely the case of the pions (π ± and π 0 ). These modes correspond to the Goldstone bosons for the symmetry breaking of × → SU L (2) SU R (2) SU I (2) . Starting from a generic lagrangian density that is invariant upon symmetry operation g ∈G and assuming canonical normalization of the pion fields, it is possible to obtain explicitly the pion-pion interaction lagrangian (density) as well as pion-nucleon interaction lagrangian: 1 r µ r 1 r r s µ r 6 Lππ = − ∂ µπ∂ π − (π ⋅ ∂ µπ)(π ⋅ ∂ π) + O(π ) 2 2F 2 = − γ µ ∂ + − ig γ µγ 5τr ⋅ ∂ πr − i γ µτr ⋅ πr × ∂ πr + LπN N( µ mN )N (N N) µ 2 (N N) ( µ ) L 2F 2F Notice that the quantities such as F and g are to be determined from the experimental input.
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